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Properties

Label 2-630-315.202-c1-0-38
Degree $2$
Conductor $630$
Sign $-0.301 + 0.953i$
Analytic cond. $5.03057$
Root an. cond. $2.24289$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.684 + 1.59i)3-s + (−0.866 − 0.499i)4-s + (0.438 − 2.19i)5-s + (−1.35 − 1.07i)6-s + (1.82 + 1.91i)7-s + (0.707 − 0.707i)8-s + (−2.06 − 2.17i)9-s + (2.00 + 0.991i)10-s + (−2.30 − 3.99i)11-s + (1.38 − 1.03i)12-s + (−4.25 + 1.14i)13-s + (−2.32 + 1.26i)14-s + (3.18 + 2.19i)15-s + (0.500 + 0.866i)16-s + (−5.57 + 5.57i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.395 + 0.918i)3-s + (−0.433 − 0.249i)4-s + (0.196 − 0.980i)5-s + (−0.554 − 0.438i)6-s + (0.690 + 0.723i)7-s + (0.249 − 0.249i)8-s + (−0.687 − 0.726i)9-s + (0.633 + 0.313i)10-s + (−0.695 − 1.20i)11-s + (0.400 − 0.298i)12-s + (−1.18 + 0.316i)13-s + (−0.620 + 0.338i)14-s + (0.823 + 0.567i)15-s + (0.125 + 0.216i)16-s + (−1.35 + 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.301 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.301 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.301 + 0.953i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.301 + 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0354976 - 0.0484483i\)
\(L(\frac12)\) \(\approx\) \(0.0354976 - 0.0484483i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (0.684 - 1.59i)T \)
5 \( 1 + (-0.438 + 2.19i)T \)
7 \( 1 + (-1.82 - 1.91i)T \)
good11 \( 1 + (2.30 + 3.99i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.25 - 1.14i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (5.57 - 5.57i)T - 17iT^{2} \)
19 \( 1 + 7.15T + 19T^{2} \)
23 \( 1 + (1.29 + 4.82i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-2.14 + 1.24i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.31 + 0.761i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.17 - 3.17i)T + 37iT^{2} \)
41 \( 1 + (-0.829 - 0.478i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.54 + 1.48i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-8.74 - 2.34i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.426 - 0.426i)T - 53iT^{2} \)
59 \( 1 + (-2.62 + 4.54i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.46 - 2.57i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.46 - 2.53i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 2.91T + 71T^{2} \)
73 \( 1 + (-1.32 - 1.32i)T + 73iT^{2} \)
79 \( 1 + (-12.2 + 7.09i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.96 + 7.34i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 3.99T + 89T^{2} \)
97 \( 1 + (5.19 + 1.39i)T + (84.0 + 48.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.36106433939192145509533768575, −9.185553477426367514294269956260, −8.573222610092662524473038194552, −8.162689152592296770164751359885, −6.36415469958605616530611829980, −5.79341989634044613108427610433, −4.76445117906759243684589802118, −4.29193308261774994949880439269, −2.29459340619603460538770450010, −0.03344374748096367456862461564, 2.01347277137488976685269570044, 2.58065419661238726833228525454, 4.38199447827784741917013314258, 5.23550385822371262524280969486, 6.74390696977365087507838495291, 7.31352773471259934144748768852, 7.951535631981476585501202790711, 9.313964904348944288808244162085, 10.35425547798053009706277910930, 10.80623925198398853639296993345

Graph of the $Z$-function along the critical line